Test objects and experimental setup

In this section we present experimental results of the serial phase retrieval with IEKF. Figure 5.6 shows the experimental setup for the layout in Figure 5.1. A 635 nm laser beam passes through a customized collimator to produce a collimated beam, and a neutral density (ND) filter is installed after the collimator to reduce the power of the beam. A test object in the beam produces the wavefront to

be estimated, which is focused by a 200 mm lens. A CCD camera with pixel size∆d = 4.54 µm is mounted on a translation stage for adjusting the defocus zk

within a 25 mm range. Lens ND filter Collimated laser beam Object CCD camera

Figure 5.6: Experimental setup for wavefront retrieval.

Figure 5.7 show the two test objects used. The first test object is a Thorlabs resolution test target R1DS1N illuminated by the collimated beam. The second test object is a laser cut aperture with acronym SIOS and a phase component produced by a lens (Thorlabs LA1464-A) with an effective focal length (EFL) of 1000mm and a back focal length (BFL) of 995.3 mm.

5.4.2

Results

The intensity measurement at the image plane using the Thorlabs resolution test target are shown in Figure 5.8. Figure 5.9 shows the phase retrieval result

(mm) Laser cut mask Thorlabs R1DS1N

Figure 5.7: Mask of the test object.

of the Thorlabs resolution test target produced by translating the camera back and forth at zk = {−12, −8, −4, 0, 4, 8, 12} mm. Figure 5.9(a) and 5.9(b) are the

amplitude and phase without using IEKF for defocus and transverse shift esti- mation. Figure 5.9(c) and 5.9(d) shows an improved reconstruction result using the IEKF. The retrieved amplitude with the IEKF is closer to the ground truth of the aperture. There appears to be a tilted wavefront in the retrieved phase. This mainly corresponds to the relative tilt between the collimated beam and the camera, and the remaining errors in transverse shift estimation also have a small effect on the detected tilted wavefront.

The intensity measurement at the image plane using the SIOS target are shown in Figure 5.10. Figure 5.11 shows the phase retrieval result of the laser cut SIOS target with a phase component produced by a convex lens. Figures 5.11(a) and 5.11(b) show the amplitude and phase result without the IEKF, and Figures 5.11(c) and 5.11(d) are those with the IEKF. The results show an obvious visual improvement when using the IEKF algorithm. Since the phase retrieved from the algorithm only returns values in the range (−π, π], an unwrapping op-

-12 mm -8 mm -4 mm 0 mm

4 mm 8 mm 12 mm

Figure 5.8: Intensity measurement of the Thorlabs resolution target at the image plane.

eration is required to remove the discontinuous 2π jumps. We use the Goldstein branch cut algorithm [96, 97, 98] for phase unwrapping.

Goldstein branch cut algorithm

The discrete formula of two-dimensional phase unwrapping can be written as

ψ(i, j) = W{φ(i, j)} = φ(i, j) + 2πt, (5.15)

where W{} is the wrapping operator, t is an integer, and ψ(i, j) and φ(i, j) are the wrapped and unwrapped phase values at pixel (i, j). We wish to determine the unwrapped phase φ(i, j) given φ(i, j) ∈ (−π, π].

We could found different answers if we follow two different paths in the phase unwrapping procedure. These inconsistencies are called residues in the

0.5 1 1.5 2 2.5 3 3.5 4 4.5 (a) -3 -2 -1 0 1 2 3 (b) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 (c) -3 -2 -1 0 1 2 3 (d)

Figure 5.9: Experimental results of the Thorlabs resolution test target. (a) and (b) are the amplitude and phase without IEKF, and (c) and (d) show the results with IEKF.

unwrapping problem [97]. The residue q can be computed by summing the phase differences around the closed path of 4 adjacent pixels

q= 4 X i=1 ∆i, (5.16) where ∆1 = W{ψ(i, j + 1) − ψ(i, j)} ∆2 = W{ψ(i + 1, j + 1) − ψ(i, j + 1)} ∆3 = W{ψ(i + 1, j) − ψ(i + 1, j + 1)} ∆4 = W{ψ(i, j) − ψ(i + 1, j)} (5.17)

The phase can be unwrapped along any path if there are no residues. Otherwise, branch cuts need be placed to balance the residues [97]. The Goldstein branch

-52 mm -48 mm -44 mm -40 mm

-36 mm -32 mm -28 mm

Figure 5.10: Intensity measurement of the SIOS target at the image plane.

cut algorithm is one of the path-following methods which generates optimal branch cuts. After identifing the residues and generating the branch cuts, the algorithm performs path integration around the branch cuts.

Figure 5.12 shows the unwrapped phase of Figure 5.11(d). Figure 5.12(a) shows the unwrapped phase in a heatmap. The pixels with zero values either have amplitude below a certain threshold or represent the branch cut pixels. Figure 5.12(b) shows the unwrapped phase in a 3D surface plot. The recovered focal length of the phase component is 997.6 mm which is within the tolerance (±1% of the BFL) provided by the manufacturer. Other factors that affect the result are the gap between the mask and the lens (<1mm) and the actual distance between the object and the 200 mm focusing lens, which may be slightly offset from the nominal focus length.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (a) -3 -2 -1 0 1 2 3 (b) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (c) -3 -2 -1 0 1 2 3 (d)

Figure 5.11: Experimental results of the laser cut mask with phase com- ponent. (a) and (b) show the result of amplitude and phase without IEKF, and (c) and (d) show the those with IEKF.

-5 0 5 x (mm) -5 0 5 y (mm) -60 -40 -20 0 Phase (rad) (a) (b)

Figure 5.12: Unwrapped phase at the pupil plane. Both (a) heatmap and (b) surface plot are included.

(a) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 (b) -3 -2 -1 0 1 2 3 (c)

Figure 5.13: Experimental results of the SIOS logo printed on a transparent paper. (a) shows the printed logo mask, and (b) and (c) show the result of amplitude and phase using IEKF.

printed logo. Figure 5.13(a) show the printed SIOS logo. Figure 5.13(b) and 5.13(c) show the result of amplitude and phase using IEKF. Both the retrieval amplitude and phase looks blurry compared to the laser cut mask. This reflects the non-homogeneous properties in the transparent material and the limited precision of the paint.

Defocus and transverse shift estimation

Figure 5.14 show the stage commands and the difference between the estimate and the command (i.e. ˆzk−P

k

i=1∆zk). The figure does not indicate the error of the

estimate since the true state is unknown, but it gives us a rough idea of the stage behavior. We can see the estimate capture the periodic pattern of the commercial stage, and the estimate indicate the overall camera position drifts slightly after 120 steps.

Figure 5.15(a) and 5.15(b) shows the estimate of ˆsx and ˆsy in subpixel. The

xaxis in the plot is the estimated camera position ˆzk. The smaller (darker) dots

0 20 40 60 80 100 120 -15 -10 -5 0 5 10 15 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15

Figure 5.14: Stage commands and the difference between the estimate ˆzk

and the command.

-10 0 10 -0.6 -0.4 -0.2 0 0.2 0.4 (a) -10 0 10 -1 0 1 2 3 4 (b)

Figure 5.15: Transverse shift estimate ˆsx and ˆsy.

can see that ˆsx varies within 1 pixel, but ˆsy has high variation and an obvious

periodic pattern. This is caused by the torque produced by the weight of the mounted camera.

These figures show that the simple model in Equation 5.7 does not fully cap- ture the behavior and dynamic of the commercial stage. However, the iterative process in the IEKF improve the convergency and return the estimates that cap-

ture part of the stage behavior. A more representative model can be used if the system parameters are known.

5.5

Conclusion

In summary, we present a novel method which applies stochastic filtering tech- niques to estimate system variables in a serial phase retrieval process. The it- erated extended Kalman filter is used for online estimation of the defocus and transverse shifts in an optical system with a moving camera. We show that the IEKF successfully improves the quality of the reconstructed wavefront and re- duces the NMSE in simulation. The proposed algorithm is used to recover the wavefront in an experiment where two different objects are tested. The IEKF ap- proach enhances the details of the reconstruction. The retrieved amplitude and phase with the IEKF appears to be more representative of the true wavefront at the pupil plane than the ones using a standard phase retrieval algorithm. We in- troduce a non-trivial phase component by adding a convex lens right before the binary mask. The estimated focal length of the lens is obtained by unwrapping the phase and the result agrees with the value given by the manufacturer.

Our method can be applied to many different optical systems that have un- known system variables, and especially when these variables can only be ob- served indirectly. Some applications replace the translation stage with a focus tunable lens to speed up the image acquisition process [43, 99]. The focal length of an electrically tunable lens is adjusted by applying a current to the actuator. Instead of using a fixed look-up table obtained offline, the focal length can be estimated online using a system model that considers the probability distribu-

tion of noise and disturbances. Some systems use a deformable mirror or dig- ital micro-mirror device (DMD) for real-time wavefront reconstruction [44, 23] where an online filtering algorithm is preferred. Many other methods that gen- erate phase diversity (e.g. oblique illuminations [100], transverse translation diversity [71]) involve uncertain variables that can also be incorporated in our online estimators.

CHAPTER 6

CONCLUSIONS

6.1

Summary

The research presented in this thesis has demonstrated advances in estimation and control of self-aligning optical systems, simultaneous system parameter es- timation in phase retrieval methods, and their application towards real-time reconstruction and alignment using focal plane sensing. The contribution are summarized below:

Automated alignment. The main purpose of the alignment task is to avoid dedicated wavefront sensors, which are expensive and produces throughput lost and non-common path error. In chapter 2 we start with a simple alignment task using image feature detection at the focal plane, and prove the automated alignment concept with a single lens and low degrees of freedom system. In chapter 4 we demonstrate an advanced self-aligning method using focal plane sensing. We utilized principle component decomposition to extract useful mea- surements at the focal plane, build nonlinear observer is the simulation, and implemented iterated extended Kalman filter and unscented Kalman filter for real time estimation and control in SIOS Optics Lab. The observability of the system is analyzed and discussed in the simulation, and a further application for reflective optical system using an off-axis parabolic mirror is demonstrated. We successfully prove that the automated alignment task can be done without adding an additional wavefront sensor as most of the current technology does, and instead use the existing camera in the system.

Phase retrieval. The goal of developing phase retrieval algorithm in this thesis is to minimize the reconstruction error caused by the error of the system vari- ables. In chapter 4 we present a novel method with combine the parallel phase retrieval framework with Bayesian filtering techniques. The well-known EM algorithm is applied in the alternating optimization process. In chapter 5 we use serial phase retrieval algorithm (multi-image phase retrieval) with Kalman filtering. The major contribution of our method is the online estimation frame- work which is preferred over a batch estimation in real time application, and the fact that the filter takes into account the model of the system, the process and measurement noise statistic. We successfully proves the improvement on the reconstructed wavefront in the experiment.

SIOS Optics Lab. All the experiments carried out in this thesis are the first set of experiments in the SIOS Optics Lab. This includes setting up the camera system, laser source, motorized stages, writing MATLAB interfaces to commu- nicate with the hardware, and selecting optical components for future use. This works accelerate the potential future experiments in the SIOS Optics Lab.